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Mirrors > Home > ILE Home > Th. List > imval | GIF version |
Description: The value of the imaginary part of a complex number. (Contributed by NM, 9-May-1999.) (Revised by Mario Carneiro, 6-Nov-2013.) |
Ref | Expression |
---|---|
imval | ⊢ (𝐴 ∈ ℂ → (ℑ‘𝐴) = (ℜ‘(𝐴 / i))) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | id 19 | . . . . 5 ⊢ (𝐴 ∈ ℂ → 𝐴 ∈ ℂ) | |
2 | ax-icn 7884 | . . . . . 6 ⊢ i ∈ ℂ | |
3 | 2 | a1i 9 | . . . . 5 ⊢ (𝐴 ∈ ℂ → i ∈ ℂ) |
4 | iap0 9118 | . . . . . 6 ⊢ i # 0 | |
5 | 4 | a1i 9 | . . . . 5 ⊢ (𝐴 ∈ ℂ → i # 0) |
6 | 1, 3, 5 | divclapd 8723 | . . . 4 ⊢ (𝐴 ∈ ℂ → (𝐴 / i) ∈ ℂ) |
7 | reval 10829 | . . . 4 ⊢ ((𝐴 / i) ∈ ℂ → (ℜ‘(𝐴 / i)) = (((𝐴 / i) + (∗‘(𝐴 / i))) / 2)) | |
8 | 6, 7 | syl 14 | . . 3 ⊢ (𝐴 ∈ ℂ → (ℜ‘(𝐴 / i)) = (((𝐴 / i) + (∗‘(𝐴 / i))) / 2)) |
9 | cjcl 10828 | . . . . . 6 ⊢ ((𝐴 / i) ∈ ℂ → (∗‘(𝐴 / i)) ∈ ℂ) | |
10 | 6, 9 | syl 14 | . . . . 5 ⊢ (𝐴 ∈ ℂ → (∗‘(𝐴 / i)) ∈ ℂ) |
11 | 6, 10 | addcld 7954 | . . . 4 ⊢ (𝐴 ∈ ℂ → ((𝐴 / i) + (∗‘(𝐴 / i))) ∈ ℂ) |
12 | 11 | halfcld 9139 | . . 3 ⊢ (𝐴 ∈ ℂ → (((𝐴 / i) + (∗‘(𝐴 / i))) / 2) ∈ ℂ) |
13 | 8, 12 | eqeltrd 2254 | . 2 ⊢ (𝐴 ∈ ℂ → (ℜ‘(𝐴 / i)) ∈ ℂ) |
14 | oveq1 5875 | . . . 4 ⊢ (𝑥 = 𝐴 → (𝑥 / i) = (𝐴 / i)) | |
15 | 14 | fveq2d 5514 | . . 3 ⊢ (𝑥 = 𝐴 → (ℜ‘(𝑥 / i)) = (ℜ‘(𝐴 / i))) |
16 | df-im 10824 | . . 3 ⊢ ℑ = (𝑥 ∈ ℂ ↦ (ℜ‘(𝑥 / i))) | |
17 | 15, 16 | fvmptg 5587 | . 2 ⊢ ((𝐴 ∈ ℂ ∧ (ℜ‘(𝐴 / i)) ∈ ℂ) → (ℑ‘𝐴) = (ℜ‘(𝐴 / i))) |
18 | 13, 17 | mpdan 421 | 1 ⊢ (𝐴 ∈ ℂ → (ℑ‘𝐴) = (ℜ‘(𝐴 / i))) |
Colors of variables: wff set class |
Syntax hints: → wi 4 = wceq 1353 ∈ wcel 2148 class class class wbr 4000 ‘cfv 5211 (class class class)co 5868 ℂcc 7787 0cc0 7789 ici 7791 + caddc 7792 # cap 8515 / cdiv 8605 2c2 8946 ∗ccj 10819 ℜcre 10820 ℑcim 10821 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-ia1 106 ax-ia2 107 ax-ia3 108 ax-in1 614 ax-in2 615 ax-io 709 ax-5 1447 ax-7 1448 ax-gen 1449 ax-ie1 1493 ax-ie2 1494 ax-8 1504 ax-10 1505 ax-11 1506 ax-i12 1507 ax-bndl 1509 ax-4 1510 ax-17 1526 ax-i9 1530 ax-ial 1534 ax-i5r 1535 ax-13 2150 ax-14 2151 ax-ext 2159 ax-sep 4118 ax-pow 4171 ax-pr 4205 ax-un 4429 ax-setind 4532 ax-cnex 7880 ax-resscn 7881 ax-1cn 7882 ax-1re 7883 ax-icn 7884 ax-addcl 7885 ax-addrcl 7886 ax-mulcl 7887 ax-mulrcl 7888 ax-addcom 7889 ax-mulcom 7890 ax-addass 7891 ax-mulass 7892 ax-distr 7893 ax-i2m1 7894 ax-0lt1 7895 ax-1rid 7896 ax-0id 7897 ax-rnegex 7898 ax-precex 7899 ax-cnre 7900 ax-pre-ltirr 7901 ax-pre-ltwlin 7902 ax-pre-lttrn 7903 ax-pre-apti 7904 ax-pre-ltadd 7905 ax-pre-mulgt0 7906 ax-pre-mulext 7907 |
This theorem depends on definitions: df-bi 117 df-3an 980 df-tru 1356 df-fal 1359 df-nf 1461 df-sb 1763 df-eu 2029 df-mo 2030 df-clab 2164 df-cleq 2170 df-clel 2173 df-nfc 2308 df-ne 2348 df-nel 2443 df-ral 2460 df-rex 2461 df-reu 2462 df-rmo 2463 df-rab 2464 df-v 2739 df-sbc 2963 df-dif 3131 df-un 3133 df-in 3135 df-ss 3142 df-pw 3576 df-sn 3597 df-pr 3598 df-op 3600 df-uni 3808 df-br 4001 df-opab 4062 df-mpt 4063 df-id 4289 df-po 4292 df-iso 4293 df-xp 4628 df-rel 4629 df-cnv 4630 df-co 4631 df-dm 4632 df-rn 4633 df-res 4634 df-ima 4635 df-iota 5173 df-fun 5213 df-fn 5214 df-f 5215 df-fv 5219 df-riota 5824 df-ov 5871 df-oprab 5872 df-mpo 5873 df-pnf 7971 df-mnf 7972 df-xr 7973 df-ltxr 7974 df-le 7975 df-sub 8107 df-neg 8108 df-reap 8509 df-ap 8516 df-div 8606 df-2 8954 df-cj 10822 df-re 10823 df-im 10824 |
This theorem is referenced by: imre 10831 reim 10832 imf 10836 crim 10838 |
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